Let \(\mathbb{C}\) be the field of complex numbers, and \(\textsf{Diff}(\mathbb{C},0)\) the group of germs of analytic diffeomorphisms at the origin. Choosing a local coordinate near the origin, any \(f \in \textsf{Diff}(\mathbb{C},0)\) is determined by a power series of coefficients \(a_i\). Taking all \(a_i \in \mathbb{R}\) we have \(\textsf{Diff}(\mathbb{R},0)\), a subgroup of \(\textsf{Diff}(\mathbb{C},0)\). The purpose of the paper under review is to prove the following
Theorem. Let \(\Gamma_g\) be the fundamental group of a closed orientable surface of genus \(g\), or of a closed non-orientable surface of genus \(g \geq 4\). Then, \(\Gamma_g\) embeds in the group \(\textsf{Diff}(\mathbb{R},0)\), and hence in \(\textsf{Diff}(\mathbb{C},0)\).
The article is divided into four parts. The first three of them provide three different proofs of the theorem.
The first proof runs through Sections 2 and 3 (for orientable surface groups) and 4 (for non-orientable surface groups). The proof uses the fact that the fundamental group of an orientable surface is fully-residually free. After proving that \(\Gamma_0\) and \(\Gamma_1\) satisfy trivially the thesis, in Theorem 3.5 it is proved that \(\Gamma_g\) embeds in \(\Gamma_2\) for \(g \geq 2\), so restricting to study the case \(g=2\), for which an explicit injective morphism \(\Gamma_2 \rightarrow \textsf{Diff}(\mathbb{R},0)\) is defined. The corresponding result for non-orientable surface groups with \(g \geq 4\) is obtained in Theorem 4.1, which splits in two cases, even and odd genus. It is also noted that the method does not apply for \(g=3\).
The second proof is based on the construction of a group topology in \(\textsf{Diff}(\mathbb{C},0)\). This is introduced in Section 5, and used in Section 6 for the proof, restricted “for simplicity” to orientable surface groups for \(g=2\), but considering a complete field \(\mathbf{k}\) instead of \(\mathbb{C}\).
Finally, a third \(p\)-adic proof is obtained in Section 7 for \(\Gamma_2\). Section 8 is devoted to some consequences of the result, and to present a couple of open questions, for instance, does there exist an embedding of \(\Gamma_2\) into the group of analytic diffeomorphisms of \(\mathbb{R}/ \mathbb{Z}\) fixing the origin? A final appendix studies free groups in \(\textsf{Diff}(\mathbb{C},0)\) and \(\textsf{Diff}(\mathbf{k},0)\).